= Solution
Split the adjacent products into two subsequences:
$$
Y_j=X_{2j-1}X_{2j},\qquad Z_j=X_{2j}X_{2j+1}\qquad(j\geq1).
$$
Within each subsequence the pairs use disjoint coordinates, so each subsequence consists of <independent and identically distributed random variables>. The two subsequences need not be <independence>[independent] of each other. By <independence> within a pair,
$$
\mathbb E|Y_j|=\mathbb E|Z_j|=(\mathbb E|X_1|)^2<\infty,\qquad
\mathbb E Y_j=\mathbb E Z_j=m^2.
$$
Apply the <strong law of large numbers> from (c) separately to obtain $k^{-1}\sum_{j=1}^kY_j\to m^2$ and $k^{-1}\sum_{j=1}^kZ_j\to m^2$ simultaneously <almost surely>. Since
$$
\frac{\widetilde S_n}{n}
=\frac{\lceil n/2\rceil}{n}\frac{\sum_{j=1}^{\lceil n/2\rceil}Y_j}{\lceil n/2\rceil}
+\frac{\lfloor n/2\rfloor}{n}\frac{\sum_{j=1}^{\lfloor n/2\rfloor}Z_j}{\lfloor n/2\rfloor},
$$
and both prefactors tend to $1/2$, the <strong law for adjacent products> gives
$$
\boxed{\frac{\widetilde S_n}{n}\longrightarrow m^2\quad\text{almost surely}.}
$$
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